Representation Theory of Compact Inverse Semigroups

Description

W. D. Munn proved that a finite dimensional representation of an inverse semigroup
is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this
thesis will be to give new analytic proof that every finite dimensional representation
of a compact inverse semigroup is equivalent to a ⋆-representation.
The second goal is to parameterize all finite dimensional irreducible representations
of a compact inverse semigroup in terms of maximal subgroups and order
theoretic properties of the idempotent set. As a consequence, we obtain a new and
simpler proof of the following theorem of Shneperman: a compact inverse semigroup
has enough finite dimensional irreducible representations to separate points if and
only if its idempotent set is totally disconnected.
Our last theorem is the following: every norm continuous irreducible ∗-representation
of a compact inverse semigroup on a Hilbert space is finite dimensional.